2. As shown in the figure, the bisector of ∠B and ∠C in △ABC passes through point P, and let ∠A = X and ∠ BPC = Y. When ∠A changes, find the functional relationship between Y and X, judge whether Y is a linear function of X, and point out that,
3. When a store sells a commodity, it increases a certain profit on the basis of the purchase price. The relationship between quantity x and sales price y is shown in the following table. Please list the functional relationship between y and x according to the information provided in the table, and find out the selling price when the quantity is 2.5 kg.
Quantity x (kg) 1 234 …
Price y (yuan) 8+0.416+0.824+1.232+1.6 …
The distance between Party A and Party B is 500 kilometers, and the car travels from Party A to Party B at a speed of 80 kilometers per hour.
(1) Write the functional relationship between the distance s (km) from the car to the second place and the departure time t (hours), and indicate whether it is a linear function;
(2) Write the range of independent variables;
(3) How long does it take for the bus to leave from A, and the distance from B100km?
I. Design of Preferential Scheme
Example 1 (Zhenjiang city) National unification. * * * In the extraordinary period of fighting against SARS, a medical instrument factory accepted the task of producing a batch of high-quality medical masks. It is required to produce 50,000 masks of type A and type B within 8 days (including 8 days), of which the number of masks of type A is not less than 1.8 million.
In this mission, the factory produced 10 thousand A-type masks. Q:
(1) The factory can make a profit of _ _ _ _ ten thousand yuan for the production of type A masks and _ _ _ ten thousand yuan for the production of type B masks;
(2) Assuming that the total profit of mask production in this factory is 10000 yuan, try to write the functional relationship about it and find out the range of independent variables;
(3) If you are the factory director:
① On the premise of completing the task, how to arrange the number of type A and type B masks to maximize the total profit? What is the maximum profit?
(2) If you want to complete the task in the shortest time, how to arrange the number of A-type and B-type masks? What's the shortest time?
Analysis: (1)0.5, 0.3 (5-);
(2) =0.5 +0.3(5- )=0.2 + 1.5,
First of all, 1.8 ≤ 5, but due to the limitation of production capacity, it is impossible to produce all type A masks within 8 days. Assuming that it takes a few days at most to produce type A, it takes (8-) days to produce type B. According to the meaning of the question, 0.6+0.8 (8-) = 5, and the solution = 7, the maximum value can only be 0.6 ×.
(3) Find the maximum value of ○1,because = 0.2+ 1.5 is a linear function and increases with the increase. When the maximum value is 4.2, take the maximum value of 0.2 × 4.2+ 1.5 = 2.32 (ten thousand yuan), that is, A 4 is generated by rows.
○2 To complete the task in the shortest time, it takes the shortest time to produce all B-types, but it needs to produce 1.8000 pieces of A-types, so except1.8000 pieces of A-types, the remaining 32000 pieces should be changed to B-types, and the shortest time required is 1.8+3.2+0.2.
Second, the design of marketing plan
Example 2 (Hubei) A newsstand orders an evening paper from a newspaper at a price of 1 yuan each. Newspapers that cannot be sold can be returned to the newspaper office at a price of 0.20 yuan. Within one month (calculated as 30 days), you can sell 100 copies every day for 20 days, and only 60 copies can be sold every day for the remaining 10 days.
(1) Write the functional relationship between them and point out the range of independent variables;
(2) How many newspapers should the newsstand order from the newspaper office every day to maximize the monthly profit? What is the maximum profit?
Analysis: (1) It is known that it should satisfy 60≤ ≤ 100. Therefore, newsstands order 30 newspapers from newspapers every month and sell (20+60× 10), with a profit of 0.3 (20+60× 10). 10 (-60) copies are returned to the newspaper office, and the loss is 0.5× 10 (-60) = 5-300 yuan, then the profit = (6+ 180)-(5-300) =+480, that is =.
The range of the independent variable is 60≤ ≤ 100, which is an integer.
(2) Because it is a linear function, and it increases with the increase, when the maximum value is 100, the maximum value is 100+480 = 580 (yuan).
Third, the design of preferential schemes.
Example 3 (Nantong City) A fruit company urgently needs to transport a batch of fruits that are not easy to store from City A to City B for sale. There are three transportation companies to choose from. The information provided by these three transportation companies is as follows:
transport
Unit transport speed (km/h) transport cost (yuan/km) packaging and handling time (hours) packaging and handling cost (yuan)
Company a 60 64 1500
Company b 5082 1000
Company C 100 103 700
Answer the following questions:
(1) If the total cost of packaging, loading and unloading and transportation of Company B and Company C is exactly twice that of Company A, find the distance between the two cities (accurate to one place);
(2) If the distance between A and B is kilometers, and the loss of this batch of fruits in the process of packaging, loading and unloading and transportation is 300 yuan/hour, which transportation company should be chosen, and the total cost paid by the fruit company (the sum of packaging, loading and unloading, transportation and loss) is the smallest?
Analysis: (1) If the distance between A and B is kilometers, the packaging, loading and unloading and transportation expenses of the three transportation companies are respectively RMB (6+ 1500) for A company, RMB (8+ 1000) for B company and RMB (10+) for C company.
(8 + 1000)+( 10 +700)=2×(6 + 1500),
Solution = 216 ≈ 217 (km);
(2) Assuming that the total expenses of Company A, Company B and Company C are respectively (unit: yuan), the time required for packaging and transportation of the three transportation companies is respectively: A (+4) hours; B (+2) hours; C (+3) hours. therefore
=6 + 1500+( +4)×300= 1 1 +2700,
=8 + 1000+( +2)×300= 14 + 1600,
= 10s+700+(+3)×300 = 13s+ 1600，
The key to choosing the company with the least cost now is to compare the size of the company.
∵ > 0, ∴ > forever established, that is to say, in company B and company C, only company C can be selected; The key to choosing a and c is to compare them with each other, which is related to the distance between a and B.
When >,11+2700 >13+1600, the solution is < 550, which shows that company C is better when the distance between the two cities is less than 550 kilometers.
When = = 550, it means that when the distance between two cities is equal to 550 kilometers, it is the same to choose company A or company C;
When it is 550, it means that when the distance between two cities exceeds 550 kilometers, it is best to choose Company A. 。
Four. Transport scheme design
There are 200 tons of chemical fertilizer in A city and 300 tons in B city. Now it is necessary to transport fertilizer to rural areas C and D. If it is transported from city A to rural areas C and D, the freight rates are 20 yuan/ton and 25 yuan/ton respectively, and the freight rates from city B to rural areas C and D are 15 yuan/ton and 22 yuan/ton respectively. Now it is known that land C needs 220 tons and land D needs 280 tons.
Analysis: According to the demand, all the fertilizers stored in cities A and B need to be shipped out, and the transportation scheme depends on the tonnage transported from one city to another. That is to say, if the tonnage is transported from city A to city C, then the remaining transportation scheme will be determined accordingly, and the required freight (yuan) is only related to the value of (ton). Therefore, the key to solve the problem is to establish the functional relationship between and.
Solution: If the total freight required to transport tons from city A to place C is RMB, then the remaining (200-) tons from city A should be transported to place D, and the remaining (220-) tons from place C should be transported from city B, that is, (220-) tons from city B to place C, and the remaining 300-(220-) tons from city B = 65438.
That is = 2+ 10060,
Because it increases with the increase, when the minimum value is taken, the value of is the smallest, 0≤ ≤200.
Therefore, when = 0, the minimum value = 10060 (yuan).
Therefore, the transportation scheme with the lowest freight cost is to transport all 200 tons from city A to place D, 220 tons from city B to place C, and the remaining 80 tons to place D. 。
Exercise questions:
1. (Hebei) A factory has 360 kilograms of raw materials A and 290 kilograms of raw materials B. It is planned to produce 50 products A and B with these two raw materials. It is known that it takes 9 kilograms of raw materials A and 3 kilograms of raw materials B to produce a product, which can make a profit in 700 yuan; To produce a type B product, 4 kg of type A raw materials and 10 kg of type B raw materials are needed, and the profit can be 1.200 yuan.
(1) What is the planned number of production pieces of products A and B? Please design it;
(2) The total profit of producing A and B products is (yuan), and the number of production pieces of a product is. Try to write the functional relationship between and, and explain with the nature of the function (1) which production scheme has the largest total profit? What is the maximum profit?
A factory in Beijing and a factory in Shanghai produce several computers at the same time. The factory in Beijing can support 65,438+00 foreign computers, and the factory in Shanghai can support 4 foreign computers. It is now decided to give Chongqing 8 sets and Hankou 6 sets. If the freight rates from Beijing to Hankou and Chongqing are 400 yuan/set and 800 yuan/set respectively, then the freight rates from Shanghai to Hankou and Chongqing are 300 yuan/set and 500 yuan/set respectively.
(1) If the total freight is 8400 yuan, how many sets will be shipped from Shanghai to Hankou?
(2) How many transportation schemes does * * * have if the total freight is required to be less than 8200 yuan?
(3) Find out the transportation scheme with the lowest total freight. What is the lowest total freight?
3. A newly-built shopping mall has three business departments: department store, clothing and household appliances, with sales staff 190. It is planned that the daily turnover of the whole shopping mall (referring to the total amount of goods sold every day) is 600,000 yuan. Due to the different nature of business, the number of sales staff assigned to the three departments will be different. According to the experience, the number of sales staff required for each commodity turnover of 65,438+100,000 yuan is shown in Table 65,438.
Table 1 Table 2
The number of people needed for commodity turnover per 10000 yuan, and the profit obtained from commodity turnover per 10000 yuan.
Department Store Category 5 Department Store Category 3,000 yuan
Clothing category 4 clothing category 500,000 yuan
Household appliances 2 Household appliances 200,000 yuan
The mall will allocate the planned daily turnover to three business departments, assuming that the turnover allocated to department stores, clothing and household appliances departments are (ten thousand yuan), (ten thousand yuan) and (ten thousand yuan) respectively (,,are all integers).
(1) Please use the included algebra to represent and z respectively;
(2) If the total daily profit of a shopping mall is expected to be (10,000 yuan) and satisfied, how should the shopping mall allocate the daily turnover to three business departments? How many salespeople should each department arrange?
The headmaster of a school will lead the "three good students" of the school to travel to Beijing in the summer vacation. A travel agency said, "If the principal buys a full ticket, the rest of the students can enjoy a half-price discount." B Travel Agency said, "All students, including the principal, enjoy a 40% discount on the full fare." If the full fare is 240 yuan.
(1) If the number of students is, travel agency A charges for A, travel agency B charges for B, and the fees of the two travel agencies are calculated separately (expression holds);
(2) When the number of students is what, the fees charged by the two travel agencies are the same;
(3) Discuss which travel agency is more beneficial to the number of students.
5. A children's wear factory has 38 meters of fabric A and 26 meters of fabric B, and now plans to produce 50 sets of L and M children's wear with these two fabrics. It is known that to make a set of L children's clothes, the fabric needs 0.5m and the fabric B 1 m, which can make a profit in 45 yuan. To make a set of M children's clothes, you need fabric A 0.9m and fabric B 0.2m, and you can make a profit. Let the number of sets of L-type children's clothes be, and the profit of producing these two kinds of children's clothes with these fabrics is (yuan).
(1) Write (meta) analytic function about (set); And find out the range of independent variables;
(2) In the production of this batch of children's wear, when the quantity of L-shaped children's wear is set, can the factory get the maximum profit? What is the maximum profit?
6. The following table shows the weight and profit of three kinds of vegetables. A car transportation company plans to transport three kinds of vegetables to other places for sale (each car is fully loaded according to regulations, and each car only contains one kind of vegetables).
Methyl ethylene propylene
The tonnage that each car can hold is 2 1 1.5.
Profit per ton of vegetables (100 yuan) 574
(1) If 8 vehicles are used to transport 1 1 ton of vegetables B and C to A for sale, how many vehicles are used to transport vegetables B and C respectively?
(2) The company plans to transport 36 tons of vegetables A, B and C in 20 cars to B for sale (no less than one car for each vegetable). How to arrange shipment to maximize the company's profit? What is the maximum profit?
If you sell a batch of goods at the beginning of the year, you can make a profit of 2000 yuan, and then deposit the principal and interest together in the bank. The bank interest is 10%. If it is sold at the end of the year, it can make a profit of 2620 yuan, but it has to pay the storage fee of 120 yuan. Are these goods sold well at the beginning of the year or at the end of the year?